Exercise 5, § 3

Let $\omega$ be an absolute value on the field $\mathbf{Q}$ of rational numbers. a) Show that $\omega$ is uniquely determined by its values at the prime numbers. b) If there is a prime number $p$ such that $\omega(p) \leq 1$, show that $\omega(q) \leq 1$ for every other prime number $q$ [find an upper bound for $\omega(q^n)$ by writing $q^n$ in the scale of $p$, and let $n \to \infty$]. c) If there exists a prime number $p$ such that $\omega(p) < 1$, show that $\omega(q) = 1$ for every other prime number $q$ [show that we cannot have $\omega(q) < 1$, by using the fact that for each integer $n > 0$ there exist two rational integers $r$ and $s$ such that $1 = rp^n + sq^n$]. Hence show that $\omega$ is then an absolute value equivalent to the $p$-adic absolute value on $\mathbf{Q}$. d) If $\omega(p) > 1$ for every prime number $p$, show that if $p$ and $q$ are any two prime numbers, then

[same method as in b)]. Hence show that in this case $\omega(x) = |x|^p$ for some $p \leq 1$. \*